determine if the given function has any points of discontinuity. explain your reasoning.\n$f(x)=\frac{x^{2}-b…

determine if the given function has any points of discontinuity. explain your reasoning.\n$f(x)=\frac{x^{2}-b^{2}}{x - b}$\nthere is a point of discontinuity at $x = b$ because the denominator has the factor $x - b$.\nthere are points of discontinuity at both $x=-b$ and $x = b$ because the numerator has factors of $x + b$ and $x - b$.\nthere is a point of discontinuity at $x=-b$ only because the factor of $x - b$ is common to both the numerator and denominator.\nthere is a point of discontinuity at $x = b$ only because the factor of $x - b$ is common to both the numerator and denominator and factors out.

determine if the given function has any points of discontinuity. explain your reasoning.\n$f(x)=\frac{x^{2}-b^{2}}{x - b}$\nthere is a point of discontinuity at $x = b$ because the denominator has the factor $x - b$.\nthere are points of discontinuity at both $x=-b$ and $x = b$ because the numerator has factors of $x + b$ and $x - b$.\nthere is a point of discontinuity at $x=-b$ only because the factor of $x - b$ is common to both the numerator and denominator.\nthere is a point of discontinuity at $x = b$ only because the factor of $x - b$ is common to both the numerator and denominator and factors out.

Answer

Answer:

There is a point of discontinuity at (x = b) because the denominator has the factor (x - b).

Explanation:

Step1: Recall discontinuity condition

A rational - function (y=\frac{f(x)}{g(x)}) is discontinuous when (g(x) = 0).

Step2: Identify the denominator

For (f(x)=\frac{x^{2}-b^{2}}{x - b}), the denominator (g(x)=x - b).

Step3: Find the zero of the denominator

Set (x - b=0), then (x = b). When (x = b), the function (f(x)=\frac{x^{2}-b^{2}}{x - b}) is undefined since division by zero is not allowed in the set of real numbers. Although (x^{2}-b^{2}=(x + b)(x - b)) and we can simplify the function to (x + b) for (x\neq b), the original function is discontinuous at (x = b) due to the form (\frac{x^{2}-b^{2}}{x - b}) having a zero - denominator at (x = b).